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3rd meeting: Divisibility and The Greatest Common Divisor 1. Divisibility Definition: If a and b are integers with a ≠ 0, we say that a divides b if there is an integer c such that b = ac. If a divides b, we say that a is factor of b. Theorem ( Division algorithm ): If a and b are integers such that b >0, then there is unique integer q and r such that a = b q + r with 0 ≤ r < b . 2. The greatest common divisor (gcd): Let a and b be given integers that are not both zero. The greatest common divisor of a nd b , denoted by gcd(a,b), is positive integer d satisfying the following: (i). d a and d b (ii). If c a and c b , then c ≤ d . Theorem: If a and b are integers that are not both zero, then there exist integers x and y such that ax + by = gcd(a,b). Two integer a and b, not both of those are zero, are said to relatively prime if gcd(a,b) =1. Theorem: Two integer a and b, not both of those are zero, are relatively prime if and only if there exist integer x and y such that ax + by = 1. Discussion: 1. Prove that a b and a c ⇒ a 2 bc . 2. Prove that if a b and c d , then ac bd . 3. Is it true that if a (b − c) , then a b or a c ? Explain your answer. 4. Is it true that if a bc , then a b or a c ? Explain your answer. 5. Is it true that if there exist x and y such that ax + by = d, then d = gcd(a,b)? Explain your answer. 6. Is it true that if (a, b) = d , then d ax + by, ∀x, y ∈ Z . Explain your answer. 7. Find integers m and n such that 45m + 75n = gcd(45, 75). 8. Find integers x and y such that 102x + 222y = gcd(102, 222). 9. Find integers k and t such that 51k + 87t = gcd(51, 87). 10. Find integers s and x such that 981s + 1234 x = gcd(981, 1234).